Symmetry, Integrability and Geometry: Methods and Applications Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells

After a summary of the TQFT wire model formalism we bridge the gap from Kuperberg equations for SU(3) spiders to Ocneanu coherence equations for systems of triangular cells on fusion graphs that describe modules associated with the fusion category of SU(3) at level k. We show how to solve these equations in a number of examples.


Foreword
Starting with the collection of irreducible integrable representations (irreps) of SU (3) at some level k (constructed in the framework of affine algebras or in the framework of quantum groups at roots of unity), the problem is to decide whether a graph encoding the action of these irreps actually defines a "healthy fusion graph" associated with a bona fide SU (3) nimrep, i.e. a module over a particular kind of fusion category. This is done by associating a complex number (a "triangular cell") with every elementary triangle of the given graph in such a way that their collection (a "self-connection") obeys a system of non trivial quadratic and quartic equations, called Ocneanu coherence equations, that can be themselves derived from another set of equations (sometimes called Kuperberg equations) describing relations between the intertwiners of the underlying fusion category. One issue is to describe and derive the coherence equations themselves. Another issue is to use them on the family of examples giving rise to SU (3) nimreps. Both problems are studied in this article.
Our paper consists of two largely independent parts. The first is a set of notes dealing with TQFT graphical models (wire models, spiders, etc.). Our motivation for this part was to show how the Ocneanu coherence equations for triangular cells of fusions graphs could be deduced from the so-called Kuperberg relations for SU (3), something that does not seem to be explained in the literature. This section was then enlarged in order to set the discussion in the larger framework of graphical TQFT models and to discuss some not so well known features that show up when comparing the SU (2) and SU (3) situations. Despite its size, this first part should not be considered as a general presentation of the subject starting from first principles, this would require a book, not an article. The second part of the paper deals about the coherence equations themselves. In particular we show, on a selection of examples chosen among quantum subgroups (module-categories) of type SU (3), how to solve them in order to obtain a self-connection on the corresponding fusion graphs.

Introduction and history
In recent years the mathematical structure of WZW models has been understood in the framework of fusion categories (also called monoidal), and more generally, in terms of modulecategories associated with a given fusion category of type G (a compact Lie group) at level k (some non-negative integer). The former can be constructed, for instance, in terms of affine Lie algebras at level k, or of quantum groups at a root of unity q that depends both on k and G. The usual notion of action of a group acting on a space is then replaced by a notion describing the action of a fusion category (say A k (G)) on another category, say E, which is not monoidal in general, and which is called a module-category. To every such action is also associated a partition function, which is modular invariant, and is characterized by a matrix acting in the vector space spanned by the simple objects of A k (G). If one chooses E = A k (G), this matrix is the identity matrix, and the theory is called diagonal. One obvious problem is to classify the possible E on which a given A k (G) can act. Two different E can sometimes be associated with the same partition function, so that classifying modular invariants and classifying module-categories of type G (something that is usually described in terms of appropriate graphs) are two different problems. In the case of A 1 = su(2) this distinction can be forgotten and the classification, which was shown to coincide with the classification of ADE Dynkin diagrams, was obtained more than twenty years ago [2]. The classification of A 2 = su(3) modular invariant partition functions was obtained by [15], and graphs that can now be interpreted as graphs describing the corresponding module-categories were discovered by [9], see also [8,4,17,6]. However, obtaining a graph describing some module action for a given fusion ring may not be enough to guarantee the existence of the category E itself. Loosely speaking, it may be that the relations existing between the simple objects of A k will be correctly represented at the level of E, but there may be also non trivial identities involving the morphisms of A k and one should check that such "relations between relations" are also correctly represented in E. For this reason, the list of candidates proposed in [9] was later slightly amended in [31]. In the case of the monoidal category A k defined by SU (3) at level k, there are two non trivial identities for the intertwiners. These identities seem to belong to the folklore and have been rediscovered several times. They were obviously used in 1993 by H. Ewen and O. Ogievetsky in their classification of quantum spaces of dimension 3 (see [14], and also [33]), but they were written down explicitly in 1996, in terms of webs, within the framework of A 2 spiders, by G. Kuperberg [23], and are sometimes called Kuperberg identities for SU (3). We shall later return to them, but here it is enough to say that they give rise, at the level of the module-categories over A k , to a set of equations between members of a particular class of 6j symbols, called triangular cells. These equations have been worked out by A. Ocneanu and presented, among other results, in a lecture given at Bariloche in January 2000. Given a candidate graph, they provide a necessary and sufficient condition for the existence of the underlying SU (3) module-category. They have been used to disregard one of the conjectured examples appearing in the slightly overcomplete list previously obtained by [9] and produce the results stated in [31]. The details, and the coherence equations themselves, were unfortunately not made available. To our knowledge, the coherence equations were first presented and discussed in Chapter 3.5 of the thesis [18] (2007), a section that was based, in parts, on a set of notes, containing the equations and studying several examples, that was written and distributed to students by one of us (R.C., reference [30] of [18]). A number of other examples was then worked out, about three years ago, by the authors, originally with no intention to publish them. Some time later, it became clear that no general presentation of the subject was going to be made available by the author of [31] and we started to work on this manuscript. We decided to incorporate a tentatively pedagogical discussion of TQFT graphical models, and stressing the differences between the SU (2) and SU (3) cases, without relying too much on the not so well-known formalism and terminology of spiders, since these graphical models provide a convenient framework to discuss Kuperberg equations together with coherence equations for Ocneanu triangular cells. Our aim was to show how to derive these equations and how to use them by selecting a class of examples. This explains the genesis of the present work, which, if not for the typing itself, was essentially finished two years ago. In the process of completing our article we became aware of a recent work by D. Evans et al. [12]: This publication uses Kuperberg equations and Ocneanu coherence equations as an input, solves them in almost all cases of type SU (3), and shows that the representations of the Hecke algebra (used in generalized RSOS models) that one obtains as a by-product from the values of the associated triangular cells, have the expected properties. This reference is quite complete and contains tables of results for all the examples that we originally planned to discuss. Nevertheless the main focus of [12] is about the corresponding representations of the Hecke algebra and associated Boltzmann weights, it is neither on graphical methods nor on the techniques used to solve the coherence equations. The present article should therefore be of independent interest, first of all because the specific aspects of SU (3) graphical models, although known by experts, do not seem to be easily available (we use these techniques to manipulate intertwiners and to derive the coherence equations, something that is not done in [12]), and also because we provide many details performing calculations leading to explicit values for the triangular cells (they agree, up to gauge freedom, with those given in [12]). In that respect, our approach and methodology often differs from the last quoted reference. Although it is not strictly necessary, some familiarity with the use of graphical models (see for instance the book [22], that mostly deals with the SU (2) situation) may help the reader. We acknowledge conversations with A. Ocneanu who introduced and studied about ten years ago (but did not make available) most of the material discussed here, often using a different terminology. The derivation of the coherence equations that we shall present in the first part of this paper is ours but it can be traced back to the oral presentation [31].

From module-categories to coherence equations
Our first ingredient is the fusion category A k defined by SU (3) at level k. It can be constructed in terms of integrable representations of an affine algebra, or in terms of particular irreducible representations of the quantum groups SU (3) q at roots of unity (q = exp(iπ/κ), with κ = k + 3). The Grothendieck ring of this monoidal category is called the fusion ring, or the Verlinde algebra. For arbitrary values of the non-negative integer k, it has two generators that are conjugate to one another, and correspond classically to the two fundamental representations of SU (3). The table of structure constants describing the multiplication of simple objects by a chosen generator is encoded by a finite size matrix of dimension r×r, with r = k(k+1)/2, which can be interpreted as the adjacency matrix of a graph (the Cayley graph of multiplication by this generator) called the fundamental fusion graph. More generally, multiplication of simple objects m × n = p N p mn p is encoded by fusion coefficients and fusion matrices N m .
The next ingredient is a category E, not necessarily monoidal, on which the previous one, A k , acts. In other words, E is a module-category over A k and its Grothendieck (abelian) group is a module over the fusion ring. The structure constants describing this action are encoded by matrices with non-negative integer coefficients, or equivalently, by oriented graphs. In the case of sl(3), the two generators are conjugated, so that the corresponding Cayley graphs, again called fusion (or quantum) graphs, just differ by orientation of edges, and one graph is enough to fully characterize the action of the fusion ring. More generally, the (module) action of a simple object m of A k on a simple object a of E, i.e. m × a = b F b ma b is encoded by "annular coefficients" and annular matrices F m .
A last ingredient to be introduced is the algebra of quantum symmetries [29,30]: given a module-category E k over a fusion category A k (G), one may define its endomorphism category O(E) = End A E which is monoidal, like A k (G), and acts on E in a way compatible with the A action. In practice one prefers to think in terms of (Grothendieck) rings and modules, and use the same notations to denote them. The ring O(E) is naturally a bimodule on A k (G), so that the simple objects m, n, . . . of the later (in particular the generators) act on O(E), in particular on its own simple objects x, y, . . . called "quantum symmetries", in two possible ways. One can therefore associate to each generator of A k (G) two matrices with non negative integer entries describing left and right multiplication, and therefore two graphs, called left and right chiral graphs whose union is a non connected graph called the Ocneanu graph. The chiral graphs associated with a generator can themselves be non connected. More generally, the structure constants describing the bimodule action are encoded by "toric matrices" W xy (write mxn = y (W xy ) mn y) that can be physically interpreted as twisted partition functions (presence of defects, see [36]). The particular matrix Z = W 00 is the usual partition function. Expressions of toric matrices for all ADE cases can be found in [40].
The category A k is modular (action of SL(2, Z)), and to every module-category E is associated, as discussed above, a quantity Z, called the modular invariant because it commutes with this action. One problem, of course, is to classify module-categories associated with A k . Very often, E is not a priori known (one does not even know what are its simple objects) but in many cases the associated modular invariant Z is known, and there exist techniques (like the modular splitting method, that we shall not discuss in this paper, see [19,7]) that allow one to "reverse the machine", i.e. to obtain the putative Grothendieck group of E and its module structure over the fusion ring, together with its fusion graph, from the knowledge of Z. As already mentioned, obtaining such a module structure over the fusion ring is usually not enough to guarantee the existence of the category E itself, unless we a priori know that it should exist (for instance in those cases of module-categories obtained by conformal embeddings).
For a module-category E of type SU (3), one can associate a complex-number, called a triangular cell, to each elementary triangle of its fusion graph. This stems from the following simple argument: in the classical theory, like in the quantum one, the tensor cube of the fundamental representation contains the identity representation. Therefore, starting from any irreducible representation of SU (3), say λ, if we tensor multiply it three times by the fundamental, we obtain a reducible representation that contains λ in its reduction. The same remark holds if λ is a simple object of a category E on which A k = A k (SU (3)) acts. It means that for any elementary triangle (a succession of three oriented edges) of the A k fusion graph, or of the graph describing the module structure of E, we have an intertwiner, but this intertwiner should be proportional to the trivial one since λ is irreducible. In other words, one associates -up to some gauge freedom -a complex number to each elementary triangle. This assignment is sometimes called 1 "a self-connection", "a connection on the system of triangular cells", or simply "a cell system". Independently of the question of determining these numbers explicitly, there are non trivial identities between them because of the existence of non trivial relations (Kuperberg identities) between the SU (3) intertwiners. One obtains two sets of equations, respectively quadratic and quartic, relating the triangular cells. These equations, that we call coherence equations for triangular cells, are sometimes nicknamed "small and large pocket equations" because of the shapes of the polygons (frames of the fusion graph) involved in their writing. Up to some gauge choice, these equations determine the values of the cells (there may be more than one solution). Such a solution can be used, in turn, to obtain a representation of the Hecke algebra.

Structure of the paper
Our aim is twofold: to explain where the coherence equations come from, and to show explicitly, in a number of cases, how these equations are used to determine explicitly the values of the triangular cells. The structure of our paper reflects this goal.
After a general discussion of graphical models that allows one to manipulate intertwining operators in a quite efficient manner (Section 3.1), and a summary of the SU (2) situation (Section 3.2), we describe the identities that are specific to the SU (3) cases, namely the quadratic and quartic Kuperberg equations between particular webs of the A 2 spider (Section 3.3). Then, by "plugging" these equations into graphs describing an SU (3) theory with boundaries, ie some chosen module-category E over A k (SU (3)) , we obtain Ocneanu equations for triangular cells (Section 3.4). Section 4 is devoted to a discussion of these equations and to examples. We take, for E, the fusion category A k itself and analyze three exceptional 2 examples, namely the three "quantum subgroups" of type SU (3) possessing self-fusion, which are known to exist at levels 5, 9 and 21. Module-categories obtained by orbifold techniques from the A k (the SU (3) analogs 3 of the D graphs), or exceptional modules without self-fusion, as well as other examples of "quantum modules" obtained using particular conjugacies, can be studied along the same lines. Finally, Subsection 4.7 is devoted to a "bad example", i.e. a graph that was a candidate for another module-category of SU (3) at level 9, but for which the coherence equations simply fail.
The reader who wants to jump directly to the coherence equations and to our description of a few specific cases may skip the beginning of our paper and starts his reading in Section 3.4, since the derivation of these equations from first principles requires a fair amount of material that will not be used in the sequel.
3 Intertwining operators and coherence equations 3.1 Generalities on graphical models (webs or wire models) From standard category theory we know that objects should be thought of as "points" and morphisms as "arrows" (connecting points). Although conceptually nice, this kind of visualization appears to be often inappropriate for explicit calculations. Graphical methods like those used in wire models, combinatorial spiders or topological quantum field theory (TQFT), happen to be more handy. A detailed presentation of such models goes beyond the scope of this article, but we nevertheless need a few concepts and ideas that we summarize below. In graphical models, diagrams are drawn on the plane, or more generally on oriented surfaces. Graphical elements of a model include a finite set of distinguished boundary points, possibly different types of wires (oriented or not) going through, and possibly different types of vertices, where the wires meet (warning: these internal vertices should not be thought of as boundary points). The general idea underlying graphical models is that boundary points represent simple objects, not arbitrary ones, in contradistinction with category theory, and that diagrams should be interpreted as morphisms connecting the boundary points (the later are supposed to be tensor multiplied). Diagrams without boundary points are just numbers. Diagrams can be formally added, or multiplied by complex numbers, and it is usually so that arbitrary diagrams can be obtained as linear sums of more basic diagrams. Diagrams can be joined or composed (in a way that is very much dependent of the chosen kind of graphical model), and this reflects the composition of morphisms. Diagrams can be read in various directions, which are equivalent, because of the existence of Frobenius isomorphisms. A given model may also include relations between some particular diagrams (identities between morphisms) and these relations may incorporate formal parameters (indeterminate). Finally, partial trace operations (tensor contraction or stitch) are defined. Considering only one type of boundary is not sufficient for the description of module-categories, so that more general graphical models also incorporate "marks", labelling the different types of boundaries.
There exist actually two variants of diagrammatical models. In the first version (the primitive or elementary version), boundary points refers to simple objects that are also generators; all other simple objects will appear when we tensor multiply the fundamental ones (that correspond classically to the fundamental representations of G). In other words all other objects are subfactors of tensor products of generators.
In the second version (the "clasp version", that we could be tempted 4 to call the "bound state version"), finite strings of consecutive fundamental boundary points are grouped together into boxes (clasps) in order to build arbitrary simple objects. This version may therefore incorporate new kinds of graphical elements, in particular new kinds of boundary points (external clasps representing arbitrary simple objects), new labels for wires, and new kinds of vertices, but one should remember that these new graphical elements should be, in principle, expressible in terms of the primitive ones.
For definiteness we suppose, from now on, that the underlying category is specified by a compact simple Lie group G and a level k (possibly infinite). When the level is finite, there is a finite number of simple objects; they are highest weight representations, denoted by a weight λ, such that λ, θ ≤ k, where θ is the highest root of G and the scalar product is defined by the fundamental quadratic form. The morphisms are intertwining operators, i.e. elements of Hom G,k spaces. The simple objects are often called "integrable irreducible representations of G at level k", a terminology that makes reference to affine algebras, but one should remember that the same category can be constructed from quantum groups at a root of unity (actually a root of −1) q = exp( iπ κ ), where κ = g + k is called altitude, and where g is the dual Coxeter number of G. Like in the classical case, simple objects can be labelled by weights or by Young tableaux (reducing to positive integers in the case G = SU (2)). If the level k is too small, not all fundamental representations of G necessarily appear, however for G = SU (N ) all the fundamental representations "exist" (or are integrable, in affine algebra parlance), as soon as k ≥ 1.
The first graphical models invented to tackle with problems of representation theory go back to the works done, almost a century ago, in mathematics by [39] and physics or quantum chemistry (recoupling theories for spin, [37]), but they have been considerably developed in more recent years by [42,20,22,24,38,23,30,11] and others. These models were mostly devoted to the study of SU (2), classical or quantum, in relation with the theory of knots or with the theory of invariants for 3-manifolds. They share many features but often differ by terminology, graphical conventions, signs, and interpretation. At the end of the 90's this type of formalism was extended beyond the SU (2) case, [41,34,23]. Combinatorial spiders, formally defined in the last reference, provide a precise terminology and nice graphical models for the study of fusion categories A k (G) associated with pairs (G, k). In wire models, diagrams are often read from top to bottom (think of this as time flow) but this is just conventional because of the existence of Frobenius maps; spiders diagrams (webs) can also be cut and read in one way or another. In wire models, two consecutive boundary vertices (or more) located on the same horizontal line (which is not drawn in general) should be tensor multiplied. The same is true, in the case of spiders, for vertices belonging to a boundary circle. Moreover the trivial representation is described by an empty drawing (the vacuum). However, as such, these graphical models are not general enough for our purposes, because although they provide combinatorial descriptions for the morphisms attached to a category like A k (SU (N )), they do not describe the collection of inter-relations existing between a monoidal category A k , a chosen module E, and the endomorphisms O(E) of the later. However, an extension of the recoupling models describing the case of SU (2) "coupled to matter" of type ADE (module-categories) was sketched in a section of [30]. The main idea was to introduce different marks (labels called AA, AE or EE) for boundary points representing simple objects of the three different types, and three kind of lines (we choose wiggle, continuous, and dashed) for their morphisms. Another idea was to choose an alternative way for drawing diagrams, by using elementary star-triangle duality. Let us consider a simple example: Take m a simple object in A, and a a simple object in E, since the former acts on the later (we have a monoidal functor from A to the endofunctors of E) we may consider vertices of type a, m, b where b is a simple object in E, or more generally we may consider "diffusions graphs" (see Fig. 1). Then, by applying a star-triangle duality transformation, we obtain in the first case a triangle with vertices marked E, A, A and edges of type AA, AE, AE. In the second case, we obtain a double triangle (a rhombus) with four edges of type AE, and a diagonal of type AA. Since the endomorphism category O of E also acts on E, we have also vertices of type a, x, b, where x is a simple object of O, and dually, triangles with edges of type EE or AE, or double triangles with four edges of type AE, and a diagonal of type EE. One can also introduce vertices of type m, n, p (for triangles AAA) since A is monoidal, and triangles EEE since O is monoidal as well. Altogether, there are four kinds of triangles and five types of tetrahedra (pairing of double-triangles) describing generalized 6j symbols. This framework seems to be general enough to handle all kinds of situations where a monoidal category A is "coupled" to a module E. Of course, for any specific example, we also have specific relations between the graphical elements of the model. The case of SU (2) and its modules was presented in [30], but we are not aware of any printed reference presenting graphical models to study SU (3) coupled to its modules. Fortunately, for the purposes of the present paper, the amount of material that we need from such a model is rather small. Although we are mainly interested in SU (3), we shall actually start with a description of the SU (2) case since it allows us to present most concepts in a simpler framework.

The fundamental version
For (SU (2), k) alone, i.e. without considering any action of the category A k on a module, and calling σ p the simple objects, we have one fundamental object σ = σ 1 corresponding classically to the two-dimensional spin 1/2 representation. There is a single non trivial intertwiner (the determinant map) which sends σ ⊗ σ to the trivial representation. This is reflected in by the fact that σ 0 appears on the r.h.s. of the equation σ 1 ⊗ σ 1 = σ 0 ⊕ σ 2 , where dim(σ p ) = [p + 1] where [n] = q n −q −n q−q −1 with q = exp(iπ/(2+k)). The corresponding projector (antisymmetric) is the Jones projector σ ⊗ σ → C ⊂ σ ⊗ σ. The Frobenius isomorphisms lead to the well known existence of an isomorphism 5 between σ and its conjugate σ. For this reason, the graphical model uses only one type of wire, which is unoriented. The fundamental intertwiner C is graphically described by the "cup diagram": read from top to bottom, it indeed tells us that σ ⊗ σ contains the trivial representation (remember that the trivial representation is not drawn). Its adjoint C † (the same diagram read bottom-up) is the "cap diagram", so that the composition of the two gives either an operator U = C † C from σ ⊗ σ to itself (the Jones projector, up to scale, displayed as a "cupcap"), or a number β = CC † displayed as a "cap-cup", i.e. a circle (closed loop). Obviously U 2 = βU . It is traditional to introduce the Jones projector e = β −1 U , so that e 2 = e and to set U 1 = U , U 2 = 1l ⊗ U , U n = 1l ⊗ · · · ⊗ 1l ⊗ U and e n = β −1 U n . A standard calculation shows that β is equal to the q-number [2]. The graphical elements of the SU (2) primitive model are therefore rather simple since we have only one relation (see Fig. 2, where the symbol D refers to any diagram) involving a single parameter, the value of the circle.

The clasp version (bound states version)
The clasp version of the A k (SU (2)) model is slightly more involved: the representation σ n = σ ⊗n is not a simple object and according to the general philosophy of graphical models, in their primitive version, being a tensor product of n simple objects, it is described by n points along a boundary line. Its endomorphism algebra (operators commuting with the action of SU (2) or SU (2) q on σ n ) is the Temperley-Lieb algebra. It is generated by linear combination of diagrams representing crossingless matching of the 2n points (n points "in" and n points "out" in the wire model, or 2n points along a circle, in the spider model), and generated, as a unital algebra, by cup-cap elements U i (pairs of U-turns in position (i, i + 1)). The standard recurrence relation for SU (2), namely σ ⊗ σ n = σ n−1 ⊕ σ n+1 (Tchebychev) shows that if n < k there exists a non-trivial intertwiner from σ ⊗ σ n to σ n+1 where σ n denotes the irreducible representation of quantum dimension [n + 1]. The corresponding projector (the Wenzl projector P n , which is symmetric) is therefore obtained as the equivariant projection of σ n to its highest weight irreducible summand σ n which is described by a clasp of size n or by a vertical line carrying 6 a label n. Its expression is given by the Wenzl recurrence formula: [n] P n−1 U n−1 P n−1 . On the r.h.s. of this equation, P n−1 is understood as P n−1 ⊗1l. The trace of P n , described by a loop carrying the label n, is µ n = Tr(P n ) = [n + 1].
The non-trivial intertwiners Y mnp : σ m ⊗ σ p → σ n are described by Y-shaped star diagrams (see Fig. 3) with composite wires carrying representation indices (SU (2) weights), or dually, by triangles, where edges are clasps. It is easy to show that the dimension of the triangle spaces is equal to 0 or 1 for all n since all matrix coefficients (N n ) mp of fusion matrices obtained from the relation N n = N n−1 N 1 − N n−2 , where N 1 is the adjacency matrix of the fusion graph A k+1 = A k are either equal to 0 or 1. When the value is not 0, the triangle (m, n, p) is called admissible. It can also be associated with an essential path of length n, from the vertex σ m to the vertex σ p , on the fusion graph (see the discussion in Section 3.2.4).
The corresponding endomorphisms P mnp : σ m ⊗ σ p → σ n ⊂ σ m ⊗ σ p are displayed as diffusion graphs of a special kind (see Fig. 3), or dually, as particular double triangles. The trace of P mnp , called theta symbol θ(m, n, p) because of its shape, is represented in Fig. 4. There are general formulae for the values of these symbols, which, for the pure SU (2) case, are symmetric in m, n, p, see for example [22]; they are obtained by decomposing Y mnp into elementary tangles or along a web basis. Composing a Y intertwiner and its adjoint in the opposite order gives a "propagator with a loop" which has to be proportional to the identity morphism of σ n since the later is irreducible, so that, by evaluating the trace on both sides, one finds the identity displayed 7 on Fig. 5. The endomorphisms µn θ(m,n,p) P mnp are projectors, as it is clear by composing the P 's vertically: P mnp P mnp = θ(m, n, p)µ −1 n P mnp . Notice the particular case θ(k − 1, 1, k) = µ k = [k + 1] so that we recover the Wenzl projectors P k = P k−1,k,1 , for instance P 2 = Y † 121 Y 121 = P 121 . As a special case, one recovers U = U 1 = Y † 101 Y 101 = P 101 . Since the triangle spaces are of dimension 1, the morphism defined by the left-hand side of Fig. 6 should be proportional to the Y-shaped intertwiner Y † qpr . This coefficient, that appears on the r.h.s. of Fig. 6, is written as the product of quantum dimensions and a new scalar quantity called the tetrahedral symbol T ET symbol (the first term). In the "pure" SU (2) theory, quantum or not, the general expression of T ET , which enjoys tetrahedral symmetry as a function of its six arguments, is known [22]. The first three arguments of T ET build an admissible triangle and the last three arguments determine skew lines (in 3-space) with respect to the first three. In the SU (2) theory, this symbol vanishes if m, n, p all refer to the fundamental representation σ because the cube σ 3 does not contain the trivial representation. This is precisely what is different in the SU (3) theory.
The (quantum) Racah and Wigner 6j symbols differ from the symbol T ET by normalizing factors. One should be warned that besides T ET , there are at least three types of quantities called "6j" symbols in the literature, even in the pure SU (2) case. The Racah symbols directly enter the structure of the weak bialgebra B associated with the data [30]: For each simple object n one introduces a vector space H n spanned by a basis whose elements are labelled by admissible triangles 8 (with fixed edge n) representing the intertwiners Y mnp ; one defines B = n B n = n End(H n ). The endomorphism product is depicted by vertical concatenation of vertical diffusion graphs 9 : H n ( pq rs )H n ′ ( r ′ s ′ tu ) = θ(rns) µn δ nn ′ δ rr ′ δ ss ′ H n ( pq tu ), where the pre-factor is  One then introduces horizontal diffusion graphs V x ( pq rs ), (equivalently, double triangles sharing a horizontal edge), by the equation displayed on the second line of Fig. 7. These new diffusion graphs build a new basis for the same algebra and define a new grading B = x B x . Using this new basis one introduces a multiplication on the dual B by concatenation of the horizontal diffusion graphs that can also be used to represent the corresponding dual basis: . Equivalently this defines a coproduct in B that can be shown to be compatible with the first product. The obtained structure is actually a finite dimensional weak Hopf algebra, a quantum groupoid. The (Racah) symbols { } used to define the pairing do not enjoy tetrahedral symmetry, but only quadrilateral symmetry. If we set x = 0 in the duality relation (second line of Fig. 7), the tetrahedron degenerates to a triangle and T ET ( p n q q 0 p ) = θ(p, n, q) that cancels one of the denominators that enters the definition of the Racah symbol, so that one obtains the closure relation Fig. 8 that will be used later. The proportionality coefficient entering this relation can be simply checked by writing p ⊗ q = n, taking the trace on both sides, the fact that the quantum dimension is a homomorphism, and using the definition of θ. One often defines (Wigner) 6j symbols by the equation The normalizing factor in the denominator involves the product of theta functions for all triangles of the given tetrahedra, so that the 6j symbols [ ] also enjoy tetrahedral symmetry. The terminology is unfortunately not standard: The classical limit of the [ ] symbols are called 6j by most physicists (and Racah 6j by Mathematica) but the { } are called 6j by [22] and [30]; one can even find, for instance in [3] and in several references studying the geometry of 3-manifolds, a definition of "unnormalized 6j symbols" that differ from T ET , and [ ] by other normalizing factors. The reader may look at [5] for relations between these quantities 10 and their use in studying the quantum groupoid structure.

Coupling of SU (2) to ADE matter
If we now couple SU (2) to ADE matter 11 , the story becomes more complicated (see our general discussion), since we have in general four types of triangles (see Fig. 9) and five types of generalized T ET , Racah, or Wigner 6j symbols coupled by pentagonal identities nicknamed "the Big Pentagon" equation (this was commented in [28], see [1]). Since E is a module-category over A k (SU (2)) we obtain endofunctors of E labelled by the simple objects σ n of A, sending objects of E to objects of E. Again, one represents diagrammatically such morphisms from a to b by Y -shaped diagrams called Y anb (see the first diagram on Fig. 10) where a and b refer to simple objects of E. By composition with the upside-down diagrams, we get the second diagram of Fig. 10 that can be traced, as in the pure SU (2) case, to get numbers θ anb , see   In particular, when n is trivial, the value of the circle associated with an irreducible object a of type E is the corresponding quantum number 12 µ a (see Fig. 12). Composing the morphisms in other directions gives the equations depicted in Fig. 13.  One has to be cautious here because the dimensionality of triangle spaces like anb can be bigger 13 than 1; in other words the space of essential paths from a to b and of length n may have a dimension bigger than 1. One should therefore sometimes introduce an extra label α for vertices such as anb. Comparing the T ET coefficients introduced in Figs. 6 and 14 one notices that m, n, p still label simple objects of A k (SU (2)), but a, b, c, replacing p, q, x, refer to simple objects of E. The other difference is that the right hand side of Fig. 14 now incorporates a summation over the extra label α, in those cases where the triangle space a, p, b has a dimension bigger than 1. Like in the pure SU (2) case one can introduce a vector space H = n H n spanned by a basis whose elements are labelled by admissible triangles (with fixed edges n standing for simple objects of A k (SU (2)) representing intertwiners Y anb . These elements can be interpreted in terms of essential paths from a to b, of length n, on the fusion graph of E (see the discussion in Section 3.2.4). In this SU (2) situation, it is known that H can also be constructed as the vector space underlying the Gelfand-Ponomarev preprojective algebra [16] associated with the corresponding ADE unoriented quiver. With k = ∞, identification stems from the fact that dimensions of the finite dimensional vector spaces H n , calculated according to both definitions, are equal (compare for instance [6] and [25]). In the case of SU (3) considered in the next section, the grading label of the horizontal space H n refers to a pair of integers n = (n 1 , n 2 ), seen as a Young tableau, and can no longer be interpreted as a length. When E = A k the ring O(E) is usually not isomorphic with A k but the construction of the bialgebra B proceeds like in the pure SU (2) case, by using vertical and horizontal diffusion graphs together with the 6j symbols appearing in Fig. 15. We shall not discuss here the structure of B, the interested reader may look at [30,1,36,5,43], but notice that these 6j symbols have four labels for the simple objects of type E (the module), one label for the type A and one label for the type O (the quantum symmetries), whereas the 6j symbols appearing in Fig. 14 have three indices of type E and three indices of type A. Using star-triangle duality, vertices become edges, AA, AE or EE, and the two kinds of 6j symbols just considered belong to the tetrahedral types described in Figs. 16, where black and white vertices refer to A and E marks. If we set x = 0 in the duality relation (second line of Fig. 15), the tetrahedron degenerates and becomes a triangle, like in  the pure SU (2) situation, and the Racah symbol . We shall need later the SU (3) analog of this closure relation, which is displayed on Fig. 17.

SU (2) braiding, Hecke and Jones-Temperley-Lieb algebras
In the SU (2) theory, one also chooses a square root A = exp( iπ 2κ ) of q and introduces a braiding 14 defined by Fig. 18. the already defined cup-cap generator. Notice that ǫX = −1/q + U . In physics, the braiding can be interpreted as a Boltzman weight at criticality in the context of RSOS height models of type SU (2). The algebra of the braid group with s + 1 strands is defined by generators g 1 , . . . , g s obeying relations g n g n+1 g n = g n+1 g n g n+1 and g n g m = g m g n when |m − n| ≥ 2. The Hecke algebra with parameter β = [2] = q + 1/q = 2 cos(π/κ) is obtained as its quotient by the ideal generated by the quadratic relation (g n + 1)(g n − q 2 ) = 0. Introducing the change of generators defined by 15 g n = q(q − U n ), the above three relations defining the Hecke algebra can be written In the SU (2) theory, the operator U is realized as the cup-cap operator appearing on the r.h.s. of Fig. 18. A simple graphical calculation shows that U n actually obeys the stronger equality U n U n±1 U n = U n , that replaces the third relation and also ensures the vanishing of a quantum Young projector defined in the algebra of the braid group with three strands (see for instance [10]). The obtained algebra is the Jones-Temperley-Lieb algebra C s β (SU (2)). In order to better understand what is modified when we trade SU (2) for SU (3), we now remind the reader how to build representations of C β (SU (2)) on vector spaces obtained from paths defined on the fusion graph of any module-category E associated with A k (SU (2)), i.e. on ADE diagrams, or on the fusion graphs of A ∞ (SU (2)), i.e. SU (2) itself and its subgroups (which are affine ADE diagrams because of the McKay correspondence).
The symbol "hat", like inξ, denotes omission. The result is therefore either 0 or a linear combination of elementary paths of length p − 2. This definition is extended by linearity to arbitrary elements of Path p a,b , called "paths". Intuitively, C n chops the round trip that possibly appears at position n. A path, not elementary in general, is called essential if it belongs to the intersection of the kernels of the annihilators C n 's (or of Jones projectors, see below). One introduces a scalar product in the vector space Path p a,b by declaring that elementary paths are orthonormal. Acting on elementary paths of length p, the creating operators C † n act as follows.
The previous sum is taken over the neighbors d of c on the fusion graph. Intuitively, this operator inserts all possible small round trip(s) in position n. The result is therefore either 0 or a linear combination of paths of length p + 2. In particular, on paths of length zero (i.e. vertices), one obtains: µa ξ ab ξ ba . The Jones' projectors e n are obtained as endomorphisms of Path p by e n . = 1 β C † n C n . All expected relations between the e n , or between the U n = βe n , are satisfied, and we obtain in this way a path realization of the Jones-Temperley-Lieb algebra, hence a representation of the Hecke algebra with parameter β. In particular, for every pair a, b of neighboring vertices: ξ ac ξ ca so that for every vertex a of the fusion graph of E, i.e. a simple object of the later, one obtains a representation on the space Path 2 a,a spanned by elementary paths of length 2 from a to a (round trips). In the SU (2) theory there are only single edges, they are un-oriented, and vertices have 1, 2 or 3 neighbors. The matrix representative of U is a direct sum of blocks of sizes (1, 1), (2,2) or (3,3). Once U is represented as above, the braiding X can be considered as one of the two fundamental connections (say 1 L ) attached to the cell system defined by E. Its value on a basic 16 cell is: The other fundamental connection (1 R ) is obtained by replacing ǫ by its conjugate ǫ in the above expression. Biunitarity of the connection is ensured by the relation ǫ 2 + ǫ 2 + β = 0. The quantity R describes a kind of generalized parallel transport from top to bottom horizontal edges along the vertical edges. The above basic cell is also a fundamental Racah symbol { a n b d x c }, for n = 1 and x = 1 L . It gives the pairing of matrix units belonging to the block of the bialgebra B labeled by n = 1, the fundamental representation of A k (SU (2)), and the block of B labelled by x = 1 L , one of the two chiral generators of O = End A k E. It can be pictured as a tetrahedron pairing a double triangle (anb)(cnd), i.e. a matrix unit of B, and a double triangle (axc)(bxd), i.e. a matrix unit of B.

Comment about the SU (2) epsilon tensor
Before embarking on SU (3) let us return to the determinant map (or epsilon tensor ǫ) of SU (2). Call {α 1 , α 2 } an orthonormal basis of the representation space associated with the fundamental σ. The intertwiner σ⊗σ → C is the epsilon tensor ǫ, defined by ǫ(α i ⊗α j ) = ǫ ij , the signature of the permutation (ij), which is 0 if i = j and ±1 otherwise. Take v, w two vectors in σ, then ǫ(v ⊗ w) = ǫ ij v i w j = v 1 w 2 − v 2 w 1 , which is the determinant det(v, w). One often says that this map σ ⊗ σ → C induces, by duality, an isomorphism from σ to σ (a particular case of Frobenius isomorphism), but there is a subtlety here: since det is antisymmetric under a circular permutation of the columns (a transposition in this case) one does not obtain a single isomorphism from σ to σ, but two, with zero sum, namely (α 1 , α 2 ) → (−α 2 , α 1 ) and (α 1 , α 2 ) → (α 2 , −α 1 ). In other words, in any wire model of SU (2) the wires will be unoriented (since σ is equivalent to its conjugate), but one should, in principle, use two distinct wires joining boundary points because there are two fundamental intertwiners. Since they have zero sum, it does not harm to make a choice once and for all, so that in practice only one kind of wire is used. The discussion is not modified when we move from SU (2) to SU (2) q . The only place where this subtlety should be remembered is when one tries to define an action of the category A k (SU (2)) on a module whose fusion graph would contain tadpoles: the presence of a tadpole -loop -at the vertex a (a simple object) would mean that, besides the identity, we could have another intertwiner from a to a, but they should be proportional since a is simple, however the sign of this proportionality factor θ(a, 1, a) will change if we select the opposite convention for the isomorphism between σ and σ. The only possible conclusion is that the proportionality factor is zero, which means that such a graph cannot be associated with any SU (2) module. This argument was presented in [31]. Another proof eliminating tadpole graphs was given in [35].

Equations for the SU (3) model
The SU (3) case shares many features with the SU (2) case. This was one of the reasons to discuss the simpler later case in the previous section, so that we can now concentrate on novel features. Besides the fact that one has to take into account orientation of edges, complex conjugation (non trivial Frobenius isomorphisms), and the fact that have usually a dimension bigger than 1, the general discussion about the structure of the weak bialgebra B and the different types of 6j symbols stays essentially the same. In particular there are again five types of tetrahedra and therefore five types of cells 17 , but those that we need to consider here are of type AAAE. As already stated, 6j of that type vanish for SU (2) when the three edges AA (simple objects of A) all coincide with the fundamental representation σ, this is precisely what is not true for SU (3).

The elementary version
For SU (3), we call {α 1 , α 2 , α 3 } the orthonormal basis of σ. Again, the intertwiner (determinant map) det : σ ⊗ σ ⊗ σ → C is defined by the epsilon tensor ǫ(α i ⊗ α j ⊗ α k ) = ǫ ijk α k , with ǫ ijk the signature of the permutation (ijk). Using Frobenius isomorphisms on the identity intertwiner σ → σ gives an intertwiner σ ⊗ σ → C represented as a cup diagram. Here σ is not equivalent to σ and the basic wire (representing an intertwiner from σ to itself, or from σ to itself) is oriented: a downward oriented wire, read from top to bottom, denotes the identity morphism from σ to σ and an upward oriented wire, also read from top to bottom, denotes the identity morphism from σ to σ. The determinant map is symmetric under circular permutation of the columns and the problem discussed for SU (2) at the end of the previous section does not arise. In particular, tadpoles in graphs describing SU (3) modules may exist. Using Frobenius isomorphisms on the determinant map σ 3 → C leads to an intertwining operator σ ⊗ σ → σ graphically represented 18 by the triple vertex ( Fig. 19) with all wires oriented in, and an intertwining operator σ ⊗ σ → σ represented by the triple vertex ( Fig. 19) with all wires oriented out. We now turn to "relations between relations" (i.e. to relations between intertwiners), there are three of them. The first, reminiscent of the SU (2) theory, says that the circle has value [3]. Indeed, in the SU (3) theory, we have the same cup-cap morphisms as before, but they are now oriented. The cap graph, read from top to bottom, says that there is an intertwiner from the trivial representation to 3 ⊗ 3, and the cup graph, read from top to bottom, says that 3 ⊗ 3 contains the trivial representation. Composition of the two, from top to bottom, gives a number, i.e. the rule displayed on Fig. 20. In the SU (2) theory there are no further relations, but in the SU (3) theory, there are two. First non-trivial relation. Consider the morphism σ → σ ⊗ σ → σ. Read from top to bottom it is represented by the left hand side of Fig. 21. This morphism goes from σ to σ, however σ is irreducible, so this morphism is proportional to the identity morphism. The equality is depicted by Fig. 21. Conventionally the proportionality constant is equal to β = q + 1/q = [2]. Tracing this equality gives immediately the value of the theta symbol: θ(σ, σ, σ) = [2] [3]. Second non-trivial relation. Let us consider the possible morphisms from σ ⊗ σ to σ ⊗ σ. There are three of them (we suppress the tensor product signs in this discussion): 1. We do nothing, i.e. σ goes to σ and σ goes to σ. It is an identity morphism depicted by the last term on the r.h.s. of Fig. 22.
2. We compose the cup and cap morphisms: σ ⊗ σ → C → σ ⊗ σ. This is depicted by the first term on the r.h.s. of Fig. 22.
3. The last possibility is to consider the chain depicted by the l.h.s. of Fig. 22: However, as we know, σ ⊗ σ = σ (1,0) ⊗ σ (0,1) = σ (0,0) ⊕ σ (1,1) , in terms of highest weights labels (or, in terms of quantum dimensions: [3] [4], or, classically, 3×3 = 1+8). So, from σ ⊗ σ we have two intertwiners, respectively to C and to σ (1,1) . By composing them with their adjoints, i.e. reversing the arrows, we have two morphisms, and only two, up to linear sums, from σ ⊗ σ to itself. This means that the three morphisms found previously cannot be independent: there is a linear relation between them. The third (for instance) can be written in terms of the first two. Actually, as displayed on Fig. 22, the third is just the sum of the first two, this was shown by [23] where it is also shown that all other relations between morphisms of SU (3) are consequences of Figs. 21 and 22. Although belonging to the folklore (we already mentioned the fact that they were explicitly used by [14]) these two relations are often called "quadratic and quartic equations for the Kuperberg SU (3) spider". The first is called quadratic because it involves two triple points and the other is quartic because it involves four triple points. Since these relations hold for SU (3), classical or quantum, they should also hold for any (classical or quantum) module, i.e. they should be true for SU (3) itself, its finite subgroups, for the fusion categories A k (SU (3)) and their module -categories E.

The clasp version (bound states version)
There is of course a clasp version of the wire models, with new types of wires for every irreducible representation existing at level k. As usual, those "bound states wires" are labelled by highest weights (p, q) or Young tableaux. By looking at the expressions of fusion matrices for SU (3) at level k, one notices that the dimension of triangle spaces can be bigger than 1, even in the pure SU (3) case, in contrast with the SU (2) theory. The endomorphism algebra of the object σ m ⊗σ n , i.e. Hom SU (3) (σ m ⊗ σ n , σ m ⊗ σ n ) is the SU (3) version of the Temperley-Lieb algebra. It can be generated by linear combination of diagrams generalizing those discussed in the SU (2) section; one can also define generalizations P mn of the Wenzl projectors, with image the irreducible representation of SU (3) indexed by the highest weight (m, n), in the endomorphism algebra of σ m ⊗ σ n (see [34,23,41]), obeying a three terms recurrence relation. One can even define a natural basis in arbitrary intertwiner spaces, see [41], however we shall not need these explicit constructions in full generality. The definition of theta symbols and 6j symbols then follows the general rules described in the SU (2) section, but this has to be done with much more care because edges are oriented, and also because the dimension d n of triangle spaces H n , where n = {n 1 , n 2 }, is usually bigger than 1, so that the notations, for example, have to incorporates new labels. The recurrence relations for SU (3) fusion matrices N n = (N q n,p ) = (N (n 1 ,n 2 ),(p 1 ,p 2 ) (q 1 ,q 2 ) ), using the seed N (1,0) , the adjacency matrix of the fusion graph, are: As usual, d n = p,q N q n,p .

Coupling to SU (3) matter: Ocneanu coherence equations for SU (3) triangular cells (self-connections)
Before deriving these equations we need to consider an extension of the SU (3) recoupling system where A k (SU (3)) is allowed to act on a module E. In a wire model, the solid wire labelled a refers to a simple object of E and the corresponding circle, labelled a denotes its Perron-Frobenius dimension µ a . A morphism from a to b, induced from the action of the simple object 19 n of A k (SU (3)) is represented, in the extended wire model, as a vertex anb, see Fig. 23 where the extra index α keeps track of multiplicities. The corresponding theta symbol is θ(a, n, b; α, α ′ ). The module action of A k (SU (3)) on E is defined by a fusion graph that will be called G in the last section: it is the Cayley graph describing the action of σ = σ (1,0) (the basic representation) on the simple objects, called a, b, . . ., of E. More generally, the action n × a = b F b na b of 19 Here n should be thought of as a multi-index since it refers to an irreducible representation of SU (3). the simple objects n of A k is described by the so-called annular matrices F n obeying the same recurrence relation as the fusion matrices N n , only the seed G = F (1,0) , is different. For any fusion graph G describing a quantum module E of type SU (3), we shall obtain two coherence equations for the system of triangular cells on the graph G. In physicists' parlance, they are obtained from the two previously discussed Kuperberg equations by coupling SU (3) to E-type matter, and taking a trace.

Normalization and closure relations
When n refers to the fundamental (and basic) representation σ = (1, 0) we usually do not put any explicit label on the corresponding oriented line of the wire model diagram, and the morphism a ⊗ σ → b is described by an oriented edge from a to b on the fusion graph of E. This intertwiner, at the moment, is only defined up to scale, so we need to fix its normalization. This is done by giving a value to the corresponding theta symbol: It is convenient to set: Fig. 24. At some point we shall need the following closure relations: Figs. 25 and 26. They can be seen as generalizations of those already obtained for SU (2) (see Figs. 8 and 17). Notice that Fig. 26 implies µ a = n (µ n /θ(a, n, a; α, α ′ ))µ a µ a δ αα ′ so that 1/µ a = n µ n /θ(a, n, a; α, α). We shall not discuss the extended recoupling theory for SU (3) in full generality, but, as already stressed, we have to associate a number to any elementary oriented triangle of the graph E. This number is a special kind of 6j symbol where three basic representations σ meet at a triple point on the wire model. It is diagrammatically denoted by the particular tetrahedral symbol T ET ( a σ b σ c σ ) displayed on Fig. 27. One should compare it with the SU (2) symbol T ET ( a p b n c m ), on the r.h.s. of Fig. 14, which is automatically 0 when all m, n, p coincide with σ. From now on we shall only consider tetrahedral symbols of that type, called triangular cells since their values T αβγ abc depend only on triangles µ α,β,γ a,b,c on the fusion graph of E, where α, β, γ denote oriented edges from a to b, b to c and c to a. In many cases there is only one edge between vertices and the notation τ (a, b, c) is sufficient.    20 Here n = σ, the fundamental representation. 21 We use a short notation for the triangle functions θ in order not to clutter the expressions. On the first line, we have replaced one tetrahedral symbol by its conjugate, using the equality displayed on Fig. 31. The sum over µ n /θ(c, n, c) on the second line was replaced by µ c −1 from the last equality obtained in Section 3.4.1. We did not write explicit labels on arcs and vertices of Fig. 30, but a summation should be carried out, not only on the arcs e but over all "dummy vertex labels", i.e. those not already labelling the vertices of Fig. 28. It is clear that the sequence of operations is easier to follow when it is displayed graphically; this is what we shall do later when we come to discuss the quartic equation.
Using θ(a, σ, c) = √ a √ c in the above, the left and right hand sides give respectively So that one obtains finally the following equality:

The quartic equation (Type II)
The calculation is similar for the quartic equation: the first step is to plug the second Kuperberg equation into a loop of type E, this is displayed on Fig. 32. On the left hand side, we use the closure equations (of Fig. 25) four times on the four sides of the square. In this way one obtains a new graph with prefactors. Then one uses the the other closure equations (see Fig. 26) on two horizontal parallel edges, together with the fact that only n = 0 contributes in the sum over intermediate states, so that the previous graph splits into two disconnected pieces. Then one uses again the closure equation of Fig. 26, twice, on the two pairs of parallel edges (again, only the term n = 0 contributes). One ends up with a product of four tetrahedral symbols, multiplied by prefactors. Those prefactors can themselves be written in terms of theta symbols, or loops. When written in terms of TQFT wires (graphs), the left hand side of the quartic equation therefore decomposes as a sum of products involving four tetrahedral symbols, four inverse "propagators" θ (which are proportional to Kronecker deltas) and one loop (a quantum dimension). This is displayed on Fig. 33. On the right hand side, the calculation is simpler. One only has to use the closure equations of Fig. 26 on each of the two terms, together with the fact that only the intermediate state n = 0 contributes. We obtain in this way a sum of two terms (Fig. 34) where each term is a product of two theta symbols. As usual, each such symbol gives a delta function and a product of square roots of quantum dimensions. After simplification, and using the equation of Fig. 31 that relates a tetrahedral symbol with its conjugate, one gets 22 : The diagrammatic interpretation of this equation, in terms of the triangles of a fusion graph, is actually simpler than what it looks. We shall return to it later. Figure 34. The right hand side of the traced quartic equation. 22 On the left hand side, one has to sum over all indices that do not appear on the right hand side.

Gauge freedom
Coherence equations of Types I and II usually do not determine completely the cell values of the triangles of a fusion graph. More precisely, given a solution, i.e. a coherent assignment of complex numbers to all triangles of the graph, one can construct other solutions by using arbitrary unitary matrices of size s ab × s ab , where s ab denote the multiplicity of edges going from vertex a to vertex b. In particular, for graphs with single edges, a gauge choice associates phase factors to edges. The effect of a gauge transformation is to multiply the values of triangular cells sharing a given edge (simple or multiple) by the chosen unitary factors. This will be illustrated later.

SU (3) braiding, Hecke and generalized JTL algebras
In the SU (3) theory, one also chooses a cubic root A = exp( iπ 3κ ) of q and introduces a braiding defined by Fig. 35. It reads X = A 2 1l − A −1 U where U now denotes the double-triple vertex The operators U n = 1l ⊗ · · · ⊗ 1l ⊗ U generate an algebra C β (SU (3)); they still obey the defining relations of the Hecke algebra with parameter β = q + q −1 (see Section 3.2.4), in particular U 2 = βU , but with an extra relation that replaces the usual SU (2) Jones-Temperley-Lieb relation. It reads: (U 3 U 2 U 1 − (U 1 + U 3 ))(U 2 U 3 U 2 − U 2 ) = 0 and can be obtained by imposing the vanishing of a quantum Young projector defined in the algebra of the braid group with four strands (see for instance [9] or [10]). Notice that C β (SU (2)) ⊂ C β (SU (3)). The operator F = U 1 U 2 U 1 − U 1 pictured on Fig. 36 is also of interest. Simple graphical calculations show that F 2 n = [2][3]F n , F n F n±1 F n = [2] 2 F n and F i F j = F j F i when |i − j| > 2, but, as noticed in [13], the operators F n do not generate an algebra isomorphic with C β (SU (2)) because the last commutation relation fails for |i − j| = 2. As with SU (2), one can represent the SU (3) intertwiners on vector spaces of paths defined on the fusion graph of any module-category E associated with A k (SU (3)), or on the fusion graphs of A ∞ (SU (3)), i.e. SU (3) itself and its subgroups. The fusion graphs are now oriented: one graph is associated with the action of σ, and another one, with opposite orientation, describes the action of σ. From now on, for definiteness, we choose the first. These graphs still obey the rigidity constraint that, in the SU (2) case, implies the condition of being simply laced, but now, there may be more than one (oriented) edge ξ ab between vertices a and b, and for this reason it is usually denoted ξ α ab . An elementary path could be defined as a succession of matching edges ξ α ab , corresponding to the action of the σ generator, but it is convenient to consider more general elementary paths containing also edges of the type ξ α ab corresponding to the action of the σ generator. The SU (2) "round trip" of length 2, that was associated with the existence of an intertwiner from σ 2 to 1l is now replaced by an elementary triangle in the fusion graph, associated with the existence of an SU (3) intertwiner C from σ 2 to σ. The action of C : Path p → Path p+1 can only be defined  on paths containing both types of edges, as it is clear from its definition, reads: Formally, the prefactor, that was µ b /µ a for a single SU (2) elementary round trip ξ ab ξ ba is now replaced, for an SU (3) elementary triangle ξ α ab ξ β bc ξ γ ca , by the prefactor T αβγ abc / √ µ a µ c . Composing C with its adjoint gives the operator U : for every integer n > 0, U n , as an endomorphism of Path p , acts as follows on elementary paths with arbitrary length, origin and extremity: The last summation runs over all edges γ such that (ξ α ab ξ β bc ξ γ ca ) and (ξ α ′ ab ′ ξ β ′ b ′ c ξ γ ca ) make a rhombus with diagonal γ, i.e. a pair of elementary triangles sharing the edge ξ γ ca from c to a. A matching pair of edges (αβ) determines three vertices a, b, c of a triangle, so that when the triangular cells T αβγ abc are known, the above equation associates to every pair (a, c) of neighboring vertices of the fusion graph an explicit square matrix whose lines and columns are labelled by triangles sharing the given two vertices. Its matrix elements associate C-numbers to the various, possibly degenerated, rhombi with given diagonal (only its endpoints are given). This representation of the Hecke algebra can sometimes be obtained by other means: there are general formula for representations associated with A k (SU (N )) itself, given in [44], and all Hecke representations associated with modules over A k (SU (3)) were obtained by "computer aided flair" in [9]. The point to be stressed is that the obtained representation is now deduced from the values of triangular cells. This was discussed in [12]. For illustration we shall only give one example of this construction (see the end of Section 4.3.2).

Triangular cells on fusion graphs
A motivation for introducing a good part of the material described so far was to explain and justify the coherence equations of Types I and II. In the coming section we shall make no explicit use of the TQFT wire models but we shall see on specific examples, i.e. on specific fusion graphs, how coherence equations can be solved. The remaining part of this article can therefore be read almost independently of the previous sections. We shall consider the following A k (SU (3)) modules: A k itself, E 5 , E 9 , E 21 . We shall also study the graph Z 9 that turns out to be rejected as describing an SU (3) module.

Self-connection on graphs
An SU (3) module E at level k is specified by its fusion graph, also called E. The adjacency matrix, called G, encodes the action of the fundamental generator σ = σ (1,0) of A k (SU (3)) on the simple objects a, b, c, . . . of E, represented as vertices. There exist a Perron-Frobenius positive measure on the set of vertices, it gives their quantum dimension denoted 23 [a] = µ a for the vertex a.
We consider the following oriented triangles of the fusion graph: where α, β, γ denote oriented edges. A cell-system T on a fusion graph G is a map from the set of oriented triangles of G to the complex numbers, that we denote as follows: The complex numbers T αβγ abc are the triangular cells of G or simply "cells" in the sequel, since they will all be of that type. As discussed previously, in order to guarantee the existence of a module-category E described by the fusion graph G, the set of triangular cells associated with this graph has to obey the following coherence equations.
(1) 23 Warning: in the previous sections, this was the notation for q-numbers but there should be no confusion. 24 It may be degenerated, it is then a single edge. 25 It may also be degenerated: see the discussion later.
Gauge equivalence. Let T be a cell system for the graph G. For any pair of vertices a, b of G, let the integer s ab denote the edge multiplicity from a to b and let U ab be a unitary matrix of size s ab × s ab . We define the complex numbers: (2) These complex numbers satisfy Type I and Type II equations, therefore T ′ is also a cell system for the graph G. The two cell systems are called gauge equivalent. An equivalence class of cell systems is called a self-connection on G. A quantity is an invariant of the cell-system if its value does not depend on the gauge choices.

Solving the cell system
The number of edges from a given vertex a to a given vertex b is equal to the matrix-element (G) ab . The number of cells of G is 1 3 a (GGG) aa = 1 3 Tr(G 3 ). These complex numbers have to satisfy Type I and Type II equations. There is one Type I equation for each pair of vertices linked by one (or more) edge(s). So the number of Type I equations is given by the number of non-zero matrix elements of G. For Type II equations, the number of frames is given by Tr(GG T GG T ) where G T is the transpose matrix of G. But the following frames (related by the two diagonal symmetries): lead to the same equations, so the number of Type II equations is reduced 26 .
The number of Type I and Type II equations is much larger than the number of cells that one has to compute. Nevertheless, these equations do not fix completely their values and one can make use of gauge freedom for that. In the examples treated, we try to choose the most convenient solution (either by imposing real values for the cells, if possible, or by making a gauge choice exhibiting the symmetry of the graph). The resolution technique for a cell system depends largely on the chosen example.

Graphs with single edges
Most fusion graphs of SU (3) have only single edges (exceptions are the A * and D 3s series, the special twisted conjugate D tc 9 and the exceptional E 9 ). Since there is no multiplicity, we can drop the edge labeling symbols and denote the cells simply by T abc . The cell values, on a given triangle, for two gauge equivalent cell systems T and T ′ will only differ by phase factors since the unitary matrices in (2) are one-dimensional in this case.
(i) Type I equations. Since α = α ′ , these equations reduce to: and only give quadratic constraints on the modulus of the cells.
(ii) Type II equations. These equations read: Doubly degenerated case (a 1 = a 3 and a 2 = a 4 ). Here both terms on the right hand side of (3) contribute. The equations read: and give quartic constraints for the modulus of the cells.
Singly degenerated cases (a 1 = a 3 or a 2 = a 4 ). Here only the first or the second term of (3) remains, and the equations read: Here also the equations only give constraints for the modulus of the cells.
Non degenerate case. Here the right hand side of (3) vanishes and the Type II equations give therefore constraints on the phase factors of the cells, once the values of their modulus have been fixed by using the previous equations.

Graphs with double edges
For these graphs, because of the possible multiplicities, the summations to be performed in Type I and II equations can be quite involved. We refer to the E 9 case treated below for an example of the discussion.
We can recover these values from the previous general formulae by taking q = 1. Indeed the coherence equations I and II hold not only for the quantum SU (3) graphs, but also for the classical ones, i.e. for the fusion graph of A ∞ and for fusion graphs (that may be called SU (3) McKay graphs) associated with all the finite subgroups of SU (3). The case of SU (3) itself is displayed on Fig. 38. The numbers inside the triangles give the squared absolute values |τ | 2 of the cells.

From triangular cells to Hecke: an example
As announced in Section 3.4.5, we show on this example how to deduce matrices representing the Hecke algebra from the obtained self-connection. Here all edges are simple, so that choosing a pair of neighboring vertices amounts to choose an edge of the fusion graph; for the same reason, there is no summation over γ in the formula for U given at the end of Section 3.4.5. Consider the edge (k, l) ← (k, l + 1). Naively there is only one rhombus with this edge as a diagonal, but there are actually four of them, i.e. four matrix elements for a 2 × 2 matrix U , since one should consider degenerated cases and also take order into account. The first degenerated rhombus (set α = α ′ = (k, l) → (k + 1, l), β = β ′ = (k + 1, l) → (k, l + 1) in the expression of U ) gives [k+1] . The two non diagonal terms (they are equal since we choose the phases to make T real) correspond to the non-degenerated case ((αβ), (α ′ , β ′ )) displayed on Fig. 39. Their value is [l+1] . To the edge (k, l) ← (k, l + 1) is therefore associated a 2 × 2 In the same way one can build two other 2 × 2 matrices, i.e. a total of three, see Fig. 39, for each of the three edges (k, l) ← (k, l + 1), (k, l) ← (k − 1, l), (k, l) ← (k+1, l−1) radiating towards the vertex (k, l). Other explicit expressions for U matrices can be found in [9] and [12].  The number of oriented triangles (and therefore the number of cells) of the graph is Tr(G)/3 = 14. There are 6 external triangles with value denoted τ i , 6 internal triangles with value denoted µ i and 2 larger internal triangles with value denoted ν 0 and ν 1 :

The E 5 graph
The symmetry of the graph suggests (but it can also be deduced from the equations) that the six cells τ i have equal modulus. This is also the case for the six cells µ i , and for the two last ones. We set: |τ i | = |τ |, |µ i | = |µ| for i = 0 to 5 and |ν 0 | = |ν 1 | = |ν|. These numbers are invariants of the cell-system. The number of Type I frames is a,b (G) ab = 24. With the above symmetry considerations, the number of independent equations associated with Type I frames is reduced to 3. The Type I frames and the corresponding equations are: We have 2 q = 2 + √ 2 and 3 q = 1 + √ 2, so we obtain: The number of Type II frames is Tr(GG T GG T ) = 108. There are 24 double degenerated frames, but considering symmetries, we end up with 3 independent equations: There are 72 singly degenerated frames, 36 of them with the same vertices on the second diagonal, they lead to the following two independent equations: The other singly degenerated frames, with the same vertices in the first diagonal, are: and lead to the same equations. We can check that the value for the modulus of the cells obtained in (4) satisfy these 5 equations. We consider now the non-degenerated case. There are 12 non-degenerated Type II frames, consisting of the following three frames: and those obtained by diagonal symmetry. They lead to the same equation: We can choose a solution by setting ν 1 = −ν 0 . Using gauge freedom (see our discussion in the next paragraph), we can make all the cells real. A solution to the cell system of E 5 is therefore given by: Fixing the phases using gauge freedom. The inner part of the fusion graphs, forgetting the six "external triangles" that have equal absolute cell values |τ |, looks like a plane projection of an octahedron, Fig. 41, seen from a direction orthogonal to one of its triangular faces, so that we The successive choices are given below, writing pairs for a chosen edge and the corresponding triangle made real and positive by that choice: (13,135), (12,123), (34,345), (56, 561), (24,234), (46, 456), (62, 612). As we have seen, equations of Type II tell us (three times) that the sum of the two pyramids whose union is the octahedron vanishes, so that ν 0 + ν 1 = 0, but gauge fixing makes ν 0 real positive, so ν 1 is both real and negative. Finally, we return to the six external triangles. For each of them, the edge common with the octahedron was already gauged fixed. This leaves us with two edges, still un-gauged, for each of them, so we can make all these cells τ real and positive. Remark: the product of cells corresponding to the 8 triangles of the octahedron is independent of the succession of gauge choices. In other words, the number is an invariant. An analogous discussion can be carried out for all examples discussed in this paper.
Fixing the modulus of the cells. We first consider Type I and Type II equations that only involve the modulus of the cells. There are 24 Type I frames, but two of them involve double edges and the corresponding equations involve the phase of the cells. Among the 22 other Type I frames, only 13 of the corresponding equations are linearly independent. The three Type I frames 0 j 1 j (and 2 j 0 j ) lead to |a j | 2 = 2 q [0 j ][1 j ] = √ 2(2 + √ 3), so that the three cells a j have equal modulus. From the three Type I frames 1 j 2 j we get 3 so that the three cells b j have the same modulus, with |b j | 2 = 2 −1/2 3 1/2 (1 + √ 3) 2 . The following 7 Type I frames give equations that involve the modulus of the 16 cells defined in (5): All other Type I frames (excepting the two involving double edges) lead to equivalent equations and give no further information. Among the Type II frames leading to equations involving the modulus of the cells, only the 6 following ones lead to new equations: At this point we have 7 + 6 = 13 linearly independent equations involving the 16 undetermined parameters that are the modulus of the cells defined in (5).
Looking for a symmetric solution. The three cells a j have the same modulus, as well as the three cells b j ; this reflects the Z 3 symmetry of the graph. We look for a solution of the cell system of E 9 reflecting this symmetry, and we define the numbers |τ | 2 and |µ| 2 by setting: With this choice, the first two equations of Type I become equivalent, and we end with only two linearly independent equations of Type I. The equations corresponding to the Type II frames defined previously read: These equations impose the symmetry condition |e 12 | = |e 21 |. The last three equations of Type II are then equivalent to the first three. Also the third equation of Type II can be written in terms of the other two and of the two equations of Type I. At this point, with the symmetric condition (6), we have 5 parameters, |τ |, |µ|, |e 11 |, |e 12 | and |e 22 | but only four linearly independent equations, so we still have one unknown parameter, say |e 22 |. Solving this set of (quadratic and quartic) equations for the modulus of the cells gives two solutions depending on this parameter: where r ± = 78 + 45 In order to determine the last parameter |e 22 |, we consider the following frames: As usual, the equations associated with these frames are obtained by summing over all vertices and edges building the pyramid of equation (1). For the frame on the r.h.s. of (8), we can only sum over the vertex 3 0 but all edges connecting the frame to 3 0 are double, so that the summation finally consists of 2 4 = 16 terms! The Type II equations corresponding to these two frames read: We can combine them in order to get rid of the factors involving phases, and we get a new equation involving only the modulus of the cells. This allows to fix the last parameter |e 22 |. For each of the two solutions in (7), we obtain two possible values for |e 22 |, namely: So one would obtain four solutions for the modulus of the e ij , with the given symmetry conditions, but actually only three remain: one of them is rejected because it leads to a negative value for |e 11 |. The two solutions with |e 22 | 2 = 0 lead to e 12 = e 21 = 0, a condition making the equations involving phases more simple, we make this choice in the following.
Fixing the phases of the cells. There are no equations involving the phases of a j and b j and we can make their values real by a gauge choice. We denote the other cells as follows: c j k = |c j k |e iθ jk , d j ℓ = |d j ℓ |e iψ jℓ , e kℓ = |e kℓ |e iφ kℓ for j = 0, 1, 2 and k, ℓ = 1, 2. The Type I frames involving double edges and the corresponding equations are: Looking for a solution with the symmetric condition (6) together with the choice |e 12 |=|e 21 |=0, the above two equations lead to the following constraints on phases: e i(ψ 11 −ψ 12 ) + e i(ψ 21 −ψ 22 ) + e i(ψ 31 −ψ 32 ) = 0.
We now consider the following Type II frames and the corresponding equations: |c j 1 | 2 e 11 e 21 + |c j 2 | 2 e 12 e 22 + c j 1 c j 2 e 12 e 21 + c j 1 c j 2 e 11 e 22 = 0.
A solution to the cell-system of E 9 . With |τ | 2 = r + , |µ| 2 = r − , |e 11 | 2 = ρ − and |e 22 | 2 = ρ + , a solution is obtained by setting φ 11 = 0, φ 22 = π, ψ j1 = θ j2 and ψ j2 = θ j1 . One can check that these cells satisfy all Type I and Type II equations and therefore give a solution to the cell-system of E 9 . Summarizing results, our solution 30 is given by: that can be written Gauge freedom and invariants for E 9 . Because of gauge freedom, the given solution for the moduli and phases of c j k , d j ℓ and e kℓ is not unique (the moduli of cells a j and b j are obviously invariant). Moreover the existence of two double lines in the graph makes the discussion of gauge freedom slightly more involved than in the other examples discussed in this paper.
Define Define c 0 = (c 0 1 , c 0 2 ), c 1 = (c 1 1 , c 1 2 ), c 2 = (c 2 1 , c 2 2 ). Gauge freedom gives c j → c j ′ = U.c j e iφ j e iψ j where U is a 2 × 2 unitary matrix (the same for all j since these pairs of cells share the same double-edge) and where φ j , ψ j are phases. One sees immediately that the following quantities are gauge invariant: The last invariant is complex and goes to its complex conjugate (also an invariant) under an odd permutations of the branches 0, 1, 2. This shows that we obtain actually two gauge nonequivalent solutions for the cell system, their existence reflects the symmetry of the given graph (keep j = 0 but permute the lower indices j = 1, j = 2 in the solution for c and d given previously).
We can also introduce vectors ). The discussion is entirely similar for the coefficients d j k (introduce a 2 × 2 unitary matrix V and U (1) phases), and leads to analogous gauge invariant quantities with the same values as above.
Together with the modules of the cells a j and b j , for j = 0, 1, 2, the above invariants provide a tool that enable one to check compatibility of seemingly different solutions. In order to illustrate this, we mention: • A solution 31 given by [12] (the authors did not directly calculated the cells for E 9 , instead they determined those of its exceptional module M 9 , an easier task, and then use the fact that the former is a Z 3 orbifold of the later): , e 11 = 0, e 22 = 0, • A solution given (without proof) by [32]: The other (gauge nonequivalent) solution is obtained by λ± ↔ λ∓ and e21 ↔ −e12.
These solutions look very different and obey all equations of Types I and II, but one can check that the gauge invariants are equal 32 .
As before we have used q-numbers identities in order to simplify the expressions. We can also write these expressions in the following way. Finally, for each of the two wings, we are left with 9 triangles on each side. Although three edges in common with the inner rectangle have already been gauged fixed on each side, there is enough gauge freedom (choice of phases) on the remaining edges to make all of the corresponding cells real and positive; for the first wing one can for instance start from 1 1 , 2 1 1 , 2 2 1 , make it real positive by the choice of a phase on the edge 1 1 , 2 1 1 and proceed towards the center while determining an appropriate sequence of independent gauge choices.
We already gave the absolute values for the cells for all possible types. Our solution for the system is obtained by taking all the cells real-valued and positive, with the exception of three cells, that should be real and negative: one of the two cells of type ρ, say the one that we called ρ ′′ , and the two cells of type σ 2 , namely those that we called σ ′ 2 and σ ′′ 2 .

A counter example: the Z 9 graph
The Z 9 is a graph that could appear at level 9, i.e. at an altitude κ = 12: q 12 = −1. We shall see that coherence equations for triangular cells cannot be satisfied in this case. The graph is displayed on Fig. 44. Vertices are denoted by 0 i and 3 j i for i, j = 1, 2, 3. The subscript i refers to the vertex triality. Quantum dimension of vertices are [0 i ] = 1 and [3 j i ] = 1/ √ 3, for i, j = 1, 2, 3. There are 22 cells for the Z 9 graph. The Z 3 symmetry of the graph suggests that the cell values should also exhibit this Z 3 symmetry. In order to prove the incompatibility of the system, we do not impose, at first, any symmetry and search for the most general solution to the cell system of Z 9 , if any.
There are 30 Type I equations for Z 9 , involving the square values of the 22 cells. These equations lead to relations between square values of the cells, and we are left with 7 linear equations of Type I involving 10 unknown square values. Among the Type II equations, 30 correspond to fully degenerated cases. Taking into account the relations obtained previously, this set reduces to 7 second degree equations involving the same 10 unknown square values. One can then check that this system of 7 linear equations and 7 second degree equations involving 10 square values of cells does not have any solution.
Let us be more specific and show explicitly that one reaches a contradiction, considering a subset of the system of equations described above involving only 7 square values of cells. We introduce the following notation for the square values of cells: for j = 1, 2, 3.
We consider the following Type I frames and the fully degenerated Type II frames associated with the same edges: The corresponding Type I and Type II equations are: The solution (b j , c j ) of this quadratic equation is We now consider the following Type I frame and the fully degenerated Type II frame associated with the same edge: The corresponding Type I and Type II equations are: We have therefore two different ways of calculating the same coefficient a. Using the solutions for b j obtained in (16) it is easy to check that the above two equations are incompatible. As claimed in [31], the graph Z 9 , that appears in the list [9], and leads to a bona fide collection of annular matrices with non-negative integer coefficients providing a representation of the fusion algebra of A 9 (SU (3)), has nevertheless to be rejected as fusion graph describing a modulecategory of type SU (3) since the coherence equations fail.

Comments
Coherence equations for others Lie groups. For the Lie groups SU (n), we have n − 1 fundamental representations σ 1 , σ 2 , . . . , σ n−1 . All of them already exist at level k = 1. Call σ 0 the trivial representation. In the case of SU (3), the main ingredient of the coherence equations is that σ ⊗3 1 contains σ 0 . This is replaced for, SU (n), n > 2, or for other Lie groups, by the fact that, for appropriate fundamental representations a, b, c, the product σ a ⊗ σ b ⊗ σ c contains σ 0 . For SU (n), one can also use the fact that the conjugate of σ p is σ p = σ n−p . What matters is that a triple point will also appear in the wire diagrams. The difference with the SU (3) case is that one has to keep track of different fundamental representations (one may introduce different kinds of colors for the wires). For the same reason the coherence equations will again involve triangular cells, but the edges of triangles, possibly of different colors, will now refer to distinct representations.